I want to devise a way to measure the initial speed of a tennis ball fired from a tennis ball cannon, but without using any speed-measuring devices. Just plan distance-measuring and physics formulas.

This question stems from a friend of mine who built a tennis ball cannon, but doesn't know how high it can go. I already know that answer if the cannon is fired straight up (so it's the maximum height of the maximum angle), which is $Y_{max} = \frac{1}{2}\frac{V_0^2}{g}$. But the flaw in that formula is you have to know how fast it's already going as it leaves the cannon.

I know of the bullet experiment, as I like to call it, where a bullet is fired into a block of wood tied to a long string. Using the energy equations, we can easily calculate the initial speed of the bullet based on the maximum height that the block of wood swings to. The result is $v_{i,bullet} = \sqrt{\frac{2y(m_{bullet} + m_{wood})}{m_{bullet}}}$ where $y$ is the maximum height of the wood and bullet.

The failure of that kind of experiment, though, when measuring a tennis ball's initial velocity is that the tennis ball is elastic. I got that formula above by equating the kinetic energy of the bullet with the potential energy of the bullet-block of wood mass, because initially the bullet has only kinetic energy but no potential energy, and at the maximum height of the bullet-wood, it has only potential energy.

With a tennis ball, it won't stick to a block of wood. So the energies there aren't as easy, and you'll still have to know some final speeds. And of course, that's hard to time manually because of how fast it will be going, and the point of this question is an experiment that you can do manually.

I was thinking about getting a large mass of something, putting it on the floor, firing at it, and then measuring the distance it moves after the ball hits it. But that stuff requires estimating the coefficient of friction to the floor (which, fair enough, can be estimated by textbooks), but it's still an elastic collision. I can't quite visualize how it will go down because the tennis ball bouncing will still have energy when it bounces off.

I could also just fire the ball straight up, wait for it to come back down, and time that amount to get $t_{max}$. I could do it several times and get a value within some good margin of error. But that sounds inelegant and, frankly, not fun at all.

Is there an experiment that we can do that will just let us measure distances things move, and from there work out the initial speed of the tennis ball? I guess the largest difficulty is it's very hard to get an inelastic collision if you shoot a tennis ball at a block of wood.

Fire it horizontally, and measure how far it goes before it hits the ground. You can calculate (or measure) how long it would take if simply dropped. That's how long it takes to go that distance. This only works if there's no spin involved. Add a fudge factor for air resistance.
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Mike DunlaveyMar 11 '13 at 17:14

"The failure of that kind of experiment, though, when measuring a tennis ball's initial velocity is that the tennis ball is inelastic." The ballistic pendulum is always inelastic. Every time. The analysis goes like this: conserve momentum in the inelastic collision between the projectile and the bob to get the initial velocity of the combined projectile--bob system; then conserve energy on the combined system to get the height of the swing. Work it backward to get the initial velocity of the projectile alone. This is the standard means of determining muzzle velocities.
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dmckee♦Mar 11 '13 at 17:33

Mike Dunlavey that's interesting. Although it needs some guesstimates on the fudge factor, and a wide open space to let the ball go all the way. It'll be on the list, though. @dmckee, I'm sorry. I meant, the tennis ball is elastic, not inelastic. I edited the question. I do know how to work it out when it's inelastic -- the trouble here being a tennis ball doesnt work that way
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markovchainMar 11 '13 at 17:50

If you use the method of measuring distances be sure to account for drag. Otherwise you will underestimate the initial velocity. Also, the drag coefficient can vary significantly throughout the flight of the tennis ball due to the presence of a "drag crisis."
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OSEMar 11 '13 at 18:02

"I do know how to work it out when it's inelastic -- the trouble here being a tennis ball doesnt work that way" You have to build some kind of capture system into the pendulum. The variation I used when I was TAing physics 101 (lab low these many years past) had a hollow for the projectile with the aperture guarded by a ratchet: balls check in, but they don't check out.
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dmckee♦Mar 11 '13 at 18:05

1 Answer
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Firing the ball straight upwards is the obvious experiment. The vertical trajectory has an analytic solution even including air resistance, so you could probably estimate the muzzle velocity and the air resistance.

If this is too boring, hammer nails into your block of wood so the ball hits the nails and is impaled on them. The experiment can be made more entertaining by putting pictures of your least favourite people on the ball :-) Suspend the block of wood from a rope, i.e. make it a pendulum, and measure the distance of the swing. Measure subsequent swings to estimate the damping. This will allow you to calculate the initial velocity of the block of wood, and therefore of the ball.